1. Field of the Invention
The present invention relates to a ball indentation tester and testing technique being used to measure the material properties when tensile test cannot be applied; welding parts with continuous property variation, brittle materials with unstable crack growth during preparation and test of specimen, and the parts in present structural use. More particularly, indentation test is non-destructive and easily applicable to obtain material properties. A new numerical indentation technique is invented by examining the finite element solutions based on the incremental plasticity theory with large geometry change. The load-depth curve from indentation test successfully converts to a stress-strain curve.
2. Background of the Related Art
While indentation test is non-destructive and easily applicable to obtain material properties, the test result is difficult to analyze because of complicated triaxial stress state under ball indenter. For this reason, the indentation test is inappropriate to measure various material properties. Thus, it is used to obtain merely hardness. Recently, however, this kind of difficulty is greatly overcome both by finite element analyses of subindenter stress and deformation fields, and by continuous measurement of load and depth. As a result, stress-strain relation can be obtained from analysis of load-depth curve.
An automated indentation test gives a stress-strain curve from measured load-depth data. FIG. 1 shows a schematic profile of indentation. Here ht and dt are ideal indentation depth and projected diameter at loaded state, and hp and dp are plastic indentation depth and projected diameter at unloaded state. With an indenter of diameter D, the following relation is delivered from spherical geometric configuration.
dt=2{square root over (htDxe2x88x92ht2)}xe2x80x83xe2x80x83(1)
Assuming that xe2x80x9cprojectedxe2x80x9d indentation diameter at loaded and unloaded states remains the same as shown in FIG. 2, Hertz expressed d (=dt=dp under this assumption) as follows.                     d        =                  2.22          ⁢                                    {                                                P                  2                                ⁢                                                                            r                      1                                        ⁢                                          r                      2                                                                                                  r                      2                                        -                                          r                      1                                                                      ⁢                                  (                                                            1                                              E                        1                                                              -                                          1                                              E                        2                                                                              )                                            }                                      1              /              3                                                          (        2        )            
where r1 and r2 are indentation radius of indenter and specimen at unloaded state, and E1 and E2 are Young""s modulus of indenter and specimen, respectively. If the indenter is rigid, r1=D/2 and r2 is a function of d and hp.
Substituting these into Eq. (3) gives:                     d        =                              [                                          0.5                ⁢                C                ⁢                                  xe2x80x83                                ⁢                D                ⁢                                  {                                                            h                      p                      2                                        +                                                                  (                                                  d                          /                          2                                                )                                            2                                                        }                                                                              h                  p                  2                                +                                                      (                                          d                      /                      2                                        )                                    2                                -                                                      h                    p                                    ⁢                  D                                                      ]                                1            /            3                                              (        3        )            
where C is 5.47P (E1xe2x88x921+E2xe2x88x921).
Tabor brought the experimental conclusion that equivalent (plastic) strain xe2x80x9cat the (Brinell and Micro Vickers) indenter contact edgexe2x80x9d is given by:                               ϵ          p                =                  0.2          ⁢                      (                          d              D                        )                                              (        4        )            
where d is calculated from Eq. (2). But, Haggag et al. ignored pile-up and sink-in of material. They simply calculated the indentation diameter d with Eq. (1) at loaded state and plastic diameter dp with Eq. (3) at unloaded state, and plastic strain with Eq. (4) by substituting dp for d.
Mean contact pressure pm is defined by pm=4P(xcfx80/d2), where P is the compressive indentation load. Then constraint factor "psgr", which is a function of equivalent plastic strain, is defined as the ratio between mean contact pressure and equivalent stress.
"psgr"(xcex5p)xe2x89xa1pm/"sgr"xe2x80x83xe2x80x83(5)
Hence, the equivalent stress is expressed in the form:                     σ        =                              4            ⁢            P                                π            ⁢                          xe2x80x83                        ⁢                          d              2                        ⁢            ψ                                              (        6        )            
Note that, in a strict sense, both equivalent plastic strain and equivalent stress are functions of location within the subindenter deformed region as well as deformation intensity itself. Thus constraint factor "psgr" is also a function of location. Francis classified the indentation states into three regions and presented the empirical formula for "psgr" with indentation test results for the various materials taken into consideration.
(1) Elastic region with recoverable deformation
(2) Transient region with elastic-plastic deformation
(3) Fully plastic region with dominant plastic deformation
Haggag et al. calculated stress-using dp instead of d in Eq. (6), and they modified Francis"" constraint factor considering that constraint factor is a function of strain rate and strain hardening.                     ψ        =                  {                                                                                        ⁢                  1.12                                                                                                ⁢                                      φ                    ≤                    1                                                                                                                                          ⁢                                      1.12                    +                    τlnφ                                                                                                                  ⁢                                      1                    ≤                    φ                    ≤                    27                                                                                                                                          ⁢                                      ψ                    max                                                                                                                  ⁢                                      φ                    ≥                    27                                                                                                          (7a)            xe2x80x83"psgr"max=2.87xcex1mxe2x80x83xe2x80x83(7b)
xcfx84xe2x89xa1("psgr"maxxe2x88x921.12)/ln 27xe2x80x83xe2x80x83(7c)
where xcex1m is constraint factor index. It is proportional to strain rate, and has the value of 1 for the material with low strain rate. By investigating the experimental results, Francis suggested a normalized variable xcfx86 in the form:                     φ        =                                            ϵ              p                        ⁢                          E              2                                            0.43            ⁢            σ                                              (        8        )            
Since equivalent strain in Eq. (4) is the value at the indenter contact edge, all the values in Eqs. (5)-(8) implicitly mean values also at the indenter contact edge.
For spherical indenter, the following relation called Meyer""s law holds between applied load P and indentation projected diameter d.
P=kdmxe2x80x83xe2x80x83(9)
where k and m are material constants when indenter diameter D is fixed, and m is Meyer""s index generally in the range of 2 to 2.5.
Meyer""s experiment reveled that index m is independent of diameter D, and k decreases with increasing D.
A=k1D1mxe2x88x922=k2D2mxe2x88x922k3D3mxe2x88x922= . . .xe2x80x83xe2x80x83(10)
where A is a constant. Substituting this into Eq. (9) gives:                               P                      d            2                          =                              A            ⁡                          (                              d                D                            )                                            m            -            2                                              (        11        )            
Equation (6) converts to Eq. (12) by Eq. (11).                     σ        =                                            4              ⁢              A                        πψ                    ⁢                                    (                              d                D                            )                                      m              -              2                                                          (        12        )            
After replacing d with dt in Eq. (11), Haggag et al. calculated yield strength "sgr"0 from the following relation of yield strength and slope A that George et al. obtained from experiment.
"sgr"0=xcex2mAxe2x80x83xe2x80x83(13)
where xcex2m is a material constant. The value of xcex2m in steel is about 0.229, which comes from analysis of tensile yield strength and A.
Rice and Rosengren proposed a stress-strain relation in piecewise power law form.                                           ϵ            t                                ϵ            o                          =                  {                                                                                        ⁢                                      σ                                          σ                      o                                                                                                                                      ⁢                                                            fo                      ⁢                                              xe2x80x83                                            ⁢                      r                      ⁢                                              xe2x80x83                                            ⁢                      σ                                        ≤                                          σ                      o                                                        ⁢                                      xe2x80x83                                                                                                                                          ⁢                                                            (                                              σ                                                  σ                          o                                                                    )                                        n                                                                                                                  ⁢                                                                                    fo                        ⁢                                                  xe2x80x83                                                ⁢                        r                        ⁢                                                  xe2x80x83                                                ⁢                        σ                                             greater than                                               σ                        o                                                              ;                                          1                       less than                       n                      ≤                      ∞                                                                                                                              (        14        )            
where "sgr"0 is yield strength, xcex50="sgr"0/E yield strain and n strain hardening exponent. Total strain xcex5t is decomposed into elastic and plastic strains (xcex5t=xcex5e+xcex5p).
FIG. 3 shows the calculation process of the material properties by Haggag""s indentation method. In the approach of Haggag et al., each repetition of loading and unloading provides one point of stress-strain data points. Thus a single indentation test usually picks up total only 6-7 data point. The approach also requires prior material constants from extra tensile tests.
The Haggag""s model for the SSM system adopts the indentation theories of Francis and Tabor established on the experimental observations and some analyses. Haggag""s approach requires prior material constants from extra tensile tests, which is one of the shortcomings.
The SSM system gives stress-strain curves through regression of load-depth data obtained from 6-7 times repetitive loading and unloading. This insufficient number of data often leads to inaccurate regression. Above all, the most critical issue in Haggag""s approach is that subindenter stress field from deformation theory is far from the real one.
In order to overcome the disadvantages as described above, the present invention provides an automated indentation system for performing a compression test by loading a compressive indentation load (P). Then, an elastic modulus (E), a yield strength ("sgr"0), and a hardening exponent (n) are calculated based on measured indentation depth (ht) and indentation load (P), and unloading slope (S).
The automated indentation system of the present invention comprises a stepmotor control system (1), measurement instrumentation (2) data acquisition system (3) and control box (4).
The stepmotor (12) is adopted for precisely controlling a traveling distance and minimizing vibration of motor.
The measurement instrumentation (2) consists of a load cell (15), a laser displacement sensor (17) for measuring the indentation depth, and a ball indenter (18).
The data acquisition system (3) includes a signal amplifier for amplifying and filtering signals received from the load cell (15) and laser displacement sensor (17).
The control box (4) is pre-stored computer programming algorisms for adjusting and controlling moving speed and direction of the stepmotor (12). It also enables to perform calculations and plot the graphs of load-depth curve or strain-stress curves according to the amplified signal data, and store and retrieve the measured signal data, material properties and produced data.
The stepmotor control system (1) comprises a cylindrical linear actuator having a ball screw (14) and backlash nut (16) for suppressing backlash, a flexible coupling (13) being connected to the ball screw (14) and stepmotor (12) for constraining the rotation and high repeatability. The stepmotor control system (1) also enables to control acceleration/deceleration of the stepmotor (12) and regulating velocity with repeatability of 3xcx9c5%.
The load cell (15) is specified based on the performance of finite element simulation of indentation test. The indentation load (P) is dependent on the ball size and material properties, and maximum indentation load is under 100 kgf for 1 mm indenter.
The laser displacement sensor for measuring indentation depth is connected parallel to a linear actuator, and measurement range of laser displacement sensor (17) is 4 mm and resolution is 0.5 xcexcm.
The ball indenter (18) is an integrated spherical indenter being made of tungsten carbide (WC) for precisely measuring an indented depth, and a diameter of indenter tip is 1 mm.
Generally, the measured indentation depth (hexp) contains an additional displacement due to system compliance (hadd). Therefore, in order to obtain an accurate indentation depth, the practical indentation depth is compensated by a displacement relationship between the measured indentation depth (hexp) and an actual indentation depth (hFEM) obtained from FEA.
A computer programming algorism is provided for performing an automated indentation test by loading a compressive indentation load (P), calculating an elastic modulus (E) and a yield strength ("sgr"0), and a hardening exponent (n) from measured indentation depth (ht) and indentation load (P), and unloading slope (S), then plotting a stress-strain curve of the indented material.
The process of computer programming algorism comprises the steps of: inputting data of measured indentation depth (ht), load (P) and unloading slope (S) from pre-stored data, computing a Young""s modulus (E) from unload slope and initially guessed values of n and xcex50, computing indentation diameters (d) from c2 equation as many as the number of load and depth data, computing equivalent plastic strains (xcex5p) and equivalent stresses ("sgr") according to the calculated indentation diameters (d), computing values of strain hardening exponent (n) and K from stress-strain relation, computing a yield stress ("sgr"0) and strain (xcex50), computing updated E, d, c2, xcex5p, "sgr", n, K, "sgr"0, and xcex50 until the updated xcex50 and n are converged within the tolerance, and outputting material properties (E, "sgr"0, n) and plotting the stress-strain curve.